Question

Find the indefinite integral ∫((x^2)/(4x^3+9)) dx

Answer 1

`\displaystyle (1)/(12) \ln( |{4x}^( 3) + 9|) + \rm C`

Explanation:

we would like to integrate the following indefinite integral:

`\displaystyle \int \frac{ {x}^(2) }{4 {x}^(3) + 9} dx`

in order to integrate it we can consider using u-substitution also known as the reverse chain rule and integration by substitution as well

we know that we can use u-substitution if the integral is in the following form

`\displaystyle \int f(g(x))g'(x)dx`

since our Integral is very close to the form we can use it

let our u and du be 4x³+9 and 12x²dx so that we can transform the Integral

as we don't have 12x² we need a little bit rearrangement

multiply both Integral and integrand by 1/12 and 12:

`\displaystyle (1)/(12) \int \frac{ 12{x}^(2) }{4 {x}^(3) + 9} dx`

apply substitution:

`\displaystyle (1)/(12) \int ( 1)/(u) du`

recall Integration rule:

`\displaystyle (1)/(12) \ln(|u|)`

back-substitute:

`\displaystyle (1)/(12) \ln( |{4x}^( 3) + 9|)`

finally we of course have to add constant of integration:

`\displaystyle (1)/(12) \ln( |{4x}^( 3) + 9|) + \rm C`

and we are done!